Motion
Related Rates
Linearization
L'Hospitals
100
No Calc
A particle moves along the x-axis. The function x(t) gives the particle's position at any time t≥0
x(t)=t4-2t2-4
What is the particle's velocity v(t)at t=1?
0
100
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3 /s. How the radius of the balloon increasing when the diameter is 50 cm. Volume of sphere is V=(4/3)pir^3
What is
1/(25\pi) (cm)/sec?
100
No Calc
The line tangent to the graph of the twice-differentiable function f at the point x=3 is used to approximate the value of f(3.25). Which of the following statements guarantees that the tangent line approximation at x=3.25 is an underestimate of f(3.25) ?
A
The function f is decreasing on the interval 3≤x≤3.25
B
The function f is increasing on the interval 3≤x≤3.25
C
The graph of the function f is concave down on the interval 3≤x≤3.25
D
The graph of the function ff is concave up on the interval 3≤x≤3.25
D
100
This is the result of
lim_(x\rightarrow1)(\sqrt(2-x)-x)/(x-1)
What is
-3/2?
200
No Calc
A particle moves along the x-axis. The graph of the particle’s velocity v(t) at time t is shown above for 0<t<4.5. How many times does the particle change direction over the time interval 0<t<4.5 ?
Two times
200
A mechanic is reboring a 6-in-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one-three thousandth of an inch every minute. How rapidly is the cylinder's volume is increasing when the bore diameter is 3.8 inches. Volume of a cylinder is V=pir^2h
What is 19π/2500 in3 per minute?
200
No Calc
The locally linear approximation of the differentiable function f at x=3 is used to approximate the value of f(3.2). The approximation at x=3.2 is an overestimate of the corresponding function value at x=3.2. Which of the following could be the graph of f?
A:
B:
C:
D:
D
200
Which of the following limits does not yield an indeterminate form?
A
limx→0 4x3/(cos(x)−1)
B
limx→3 ln(x/3)/(x2−7x+12)
C
limx→π (π−x)/(sin(2x)−1)
D
limx→∞ x10/(e2x+x)
C
300
Let the function
P(t)=(3e^-sin(t))/t
represent the amount of medication (in mg) in a patient's system at t minutes. What is the rate at which the medication is changing at 6 minutes rounded to the nearest hundredth.
What is
-0.75 (mg)/(min)?
300
A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2 m3 /min, What is the rate at which the water level is rising when the water is 3 m deep. Volume of cone is V=(1/3)pir^2h
What is 8/9π m per min?
300
Let f(x) be a differentiable function that is concave up across its domain. This is the approximate value of f(3.4) at x=3 given f(3)=9 and f'(3)=5.
What is
f(3.4)\approx11?
300
This is the result of
lim_(x\rightarrow0)tan(x)/(x+sin(x))
What is
1/2?
400
No Calc
A particle moves along the y-axis so that at time t≥0 its position is given by y(t)=(2/3)t3-5t2+8t. Over the time interval 0<t<5, for what values of t is the speed of the particle increasing?
(1,2.5)U(4,5)
400
A piece of rubber tubing maintains a cylindrical shape as it is stretched. At the instant that the inner radius of the tube is 2 milimeters and the height is 20 milimeters, the inner radius is decreasing at the rate of 0.1 milimeter per second and the height is increasing at the rate of 3 milimeters per second. Which of the following statements about the volume of the tube ist rue at this instant? (Volume of cylinder is V=pir^2h)
A) The volume is increasing by 4pi cubic milimeters per second.
B) The volume is decreasing by 4pi cubic milimeters per second.
C) The volume is decreasing by 20pi cubic milimeters per second.
D) The volume is increasing by 20pi cubmic milimeters per second.
A
400
No Calc
Let f be the function given by f(x) = 2 cos x + 1. What is the approximation for f(1.5) found by using the line tangent to the graph of f at x = π/2 ?
π-2
400
No Calc
The figure above shows the graph of the twice-differentiable function f and the line tangent to the graph of f at the point (0,2). The value of
limx→0 (f(x)e-x−2)/(x2−2x) is
2
500
The function g(t) models the number of books sold in a store on day t, where t is the number of days after January 1, 2020. Which of the following is the best interpretation of the statement g'(7)=-11?
A) On January 8, 2020, approximately 11 books were sold.
B) On January 8, 2020, the number of books sold was decreasing at a rate of 11 books per day.
C) On January 8, 2020, the rate at which the books were sold was decreasing at a rate of 11 books per day per day.
D) From Januaray 1, 2020, to January 8, 2020, the number of book sold was decreasing at an average rate of 11 books per day.
B
500
The area of a circle is increasing at a rate of 213 square feet per second. At the time when the area of the circle is 16π, what is the rate of change of the circumference of the circle? Round your answer to three decimal places (if necessary).
53.250 ft/sec
500
Let f(x) = sin(x). What is the approximate value of sin(2.7) at x = 5π / 6 using linearization rounded to the nearest hundredth.
0.43
500
This is the result of
lim_(x\rightarrow\infty)ln(1+e^x)/(5x)
What is
1/5?